High Frequency Trading and Fragility

Transcription

1 High Frequency Trading and Fragility Giovanni Cespa and Xavier Vives September 206 Abstract We show that limited dealer participation in the market, coupled with an informational friction resulting from high frequency trading, can induce strategic complementarities between liquidity consumption and provision: traders consume more liquidity when the cost of liquidity provision increases, which in turn jacks up the cost of liquidity provision. This can generate market instability, where an initial dearth of liquidity degenerates into a liquidity rout (as in a flash crash). While in a transparent market, liquidity is increasing in the proportion of high frequency traders, in an opaque market strategic complementarities can make liquidity U-shaped in this proportion. Keywords: Market fragmentation, high frequency trading, flash crash, asymmetric information. JEL Classification Numbers: G0, G2, G4 A previous version of this paper circulated with the tile The welfare impact of high frequency trading. For helpful comments we thank Bruno Biais, Evangelos Benos, Thierry Foucault, Denis Gromb, Pete Kyle, Albert Menkveld, Sophie Moinas, Andreas Park, Joël Peress, Liyan Yang, Bart Yueshen, and seminar participants at INSEAD, HEC (Paris), Rotterdam School of Management, Stockholm School of Economics, the 9th Annual Central Bank Workshop on Microstructure (Frankfurt, 9/3), the conference on High Frequency Trading at Imperial College, Brevan Howard Centre (London, 2/4), the Workshop on Microstructure Theory and Applications (Cambridge, 3/5), the third workshop on Information Frictions and Learning (Barcelona, 6/5), the Bank of England, and the AFA (San Francisco, /6). Cespa acknowledges financial support from the Bank of England (grant no. RDC539). This paper has been prepared by Vives under the Wim Duisenberg Research Fellowship Program sponsored by the ECB. Any views expressed are only those of the authors and do not necessarily represent the views of the ECB or the Eurosystem. Cass Business School, City University London, CSEF, and CEPR. 06, Bunhill Row, London ECY 8TZ, UK. giovanni.cespa@gmail.com IESE Business School, Avinguda Pearson, Barcelona, Spain.

2 The report describes how on October 5, some algos pulled back by widening their spreads and other reduced the size of their trading interest. Whether such dynamic can further increase volatility in an already volatile period is a question worth asking, but a difficult one to answer. (Remarks Before the Conference on the Evolving Structure of the U.S. Treasury Market (Oct. 2, 205), Timothy Massad, Chairman, CFTC.) Introduction Concern for crashes has recently revived, in the wake of the sizeable number of flash events that have affected different markets. For futures, in the 5-year period from 200, more than a 00 flash events have occurred (see Figure ). For other contracts, the list of events where markets suddenly crash and recover is by now quite extensive. Hourly Flash Events for Selected Contracts A common trait of these episodes seems to be the apparent jamming of the rationing function of market illiquidity. Indeed, in normal market conditions, traders perceive a lack of liquidity as a cost, which in turn leads them to limit their demand for immediacy (i.e., the demand for liquidity is downward sloping). 2 This eases the pressure on liquidity suppliers, thereby producing a stabilizing effect on the market. However, on occasions a bout of illiquidity no longer has a stabilizing impact, and can instead foster a disorderly run for the exit that is conducive to a rout. In this case traders attempt to place orders despite the liquidity shortage, and demand for liquidity may be upward sloping. In these conditions, liquidity is fragile. What can account for such a dualistic feature of market illiquidity? Starting with the May 6, 200 U.S. flash-crash where U.S. equity indices dropped by 5-6% and recovered within half an hour; moving to the October 5, 204 Treasury Bond crash, where the yield on the benchmark 0-year U.S. government bond, dipped 33 basis points to.86% and reversed to 2.3% by the end of the trading day; to end with the August 25, 205 ETF market freeze, during which more than a fifth of all U.S.-listed exchange traded funds and products were forced to stop trading. More evidence of flash events is provided by NANEX. 2 To minimize market impact and the associated trading costs they incur (e.g., by using algorithms that parcel out their orders). 2

3 In this paper, we argue that important ingredients in the answer to this question are represented by the fragmentation of liquidity supply and the informational frictions induced by computerized trading. 3 Indeed, the automation of the trading process fosters liquidity supply fragmentation by limiting the market participation of some liquidity suppliers (Duffie (200) and SEC (200)); at the same time, computerized trading creates informational frictions by hampering some traders access to reliable and timely market information (Ding et al. (204)). 4 The message of the paper in a nutshell is as follows. In a dynamic market where liquidity traders have a short horizon and some dealers/market makers are not always present in the market (because this is costly), the market is stable and increasing the proportion of dealers with full market participation is good for liquidity. Suppose now that, due to a technological development, trading can be made more frequent but at the cost of introducing an informational friction (with some of the liquidity traders not able to know precisely the state of market imbalances in previous periods). Then the market may be unstable, and increasing the proportion of full dealers may be bad for liquidity. We analyze a model in which two classes of risk-averse dealers provide liquidity to two cohorts of risk-averse, short-term traders who receive a common endowment shock, in a twoperiod market. Traders enter the market to partially hedge their exposure to the risky asset. In the first round of trade both dealers types absorb the (market) orders of the first traders cohort. In the second trading round, only one class of dealers, named full, is able to participate. Full dealers, like stylized high frequency traders (HFTs), are continuously in the market and can therefore accommodate the reverting orders of the first traders cohort, as well as those of the incoming second cohort who observe an imperfect signal about the first period order imbalance. 5 A central finding of our analysis is that dealers limited market participation favors the propagation of the endowment shock across time. This is because when first period traders load their positions, a part of their orders is absorbed by standard dealers. These agents, however, are not in the market in the second period, when first period traders unwind. As a consequence, an order imbalance (induced by first period traders unwinding orders and) affecting the second period price, arises. As standard dealers are unable to rebalance in the second period, they require a larger price concession to absorb traders orders. This implies that if liquidity dries up, standard dealers absorb more of the imbalance, magnifying the propagation effect. We first study a benchmark market in which second period traders have access to a perfect 3 Automated trading is by now pervasive across different markets. For financial futures, automated trading accounts for about two-thirds of the activity in Eurodollars and Treasury contracts (Source: Keynote Address of CFTC Commissioner J. Christopher Giancarlo before the 205 ISDA Annual Asia Pacific Conference). 4 Ding et al. (204) argue that in the U.S....not all market participants have equal access to trade and quote information. Both physical proximity to the exchange and the technology of the trading system contribute to the latency. 5 In a companion paper, we then embed the baseline model in a simple platform competition setup in which exchanges compete in the supply of trading services (co-location capacity). In this framework we endogenize the decision of a dealer to acquire the technology to be continuously in the market, and the number of exchanges supplying trading services. 3

4 signal on the first period imbalance. This situation is likely to arise at low trading frequencies (e.g., intradaily), or in a transparent setup where all market participants have access to the same type of feed, even at high frequencies. In this case we show that first period traders demand for liquidity is a decreasing function of illiquidity (i.e., the compensation that dealers demand to hold the asset inventory in equilibrium): the less liquid is the market, the higher is the cost these traders incur to reduce exposure, and the less aggressive is their liquidity consumption (the closer to zero is their hedging aggressiveness). Furthermore, illiquidity is increasing in traders hedging aggressiveness (the inverse supply for liquidity is upward sloping). This is because lower aggressiveness limits liquidity consumption, which in turn shrinks dealers inventory, allowing for cheaper liquidity provision. Thus, illiquidity in this case has a direct, rationing effect on traders liquidity consumption, and a unique equilibrium arises. Furthermore, along this equilibrium, small shocks to the model s parameters have a minimal impact on market liquidity. In contrast, when access to imbalance information is impaired, the market is opaque, and illiquidity also displays a feedback, liquidity consumption expanding effect. This can create a self-sustaining loop where a liquidity evaporation, short of curtailing traders liquidity demand, fosters a stronger liquidity consumption. As a consequence, the demand for liquidity can become increasing in illiquidity, and multiple equilibria can arise. To see this, note that due to endowment shock propagation, second period traders speculate against the imbalance generated by their first period peers the more, the stronger is such propagation. Suppose now that liquidity evaporates in the first period market. As a consequence, standard dealers intermediate more of the outstanding imbalance, magnifying the propagation of the first period endowment shock, and leading second period traders to trade more aggressively against it. However, as information on the first period imbalance is noisy, these trades increase the first period uncertainty about the second period price. This can lead first period traders to consume more liquidity (as holding exposure to the asset becomes riskier), and liquidity suppliers to charge more to absorb the order imbalance (as their inventory of the risky asset increases), eventually reinforcing the initial, negative shock to market liquidity. Equilibrium multiplicity induces three levels of liquidity that can be ranked in an increasing order (low, intermediate, and high). At the low (respectively, intermediate, and high) liquidity equilibrium, volatility and liquidity consumption are high (respectively, intermediate, and low). Thus, our paper highlights a channel through which the combined effect of a heightened demand for liquidity, and a reduced liquidity provision conjure to increase market volatility, providing a positive answer to this paper s opening quotation. The liquidity consumption ranking across equilibria is a further manifestation of the fact that opaqueness jams the direct, rationing effect of illiquidity, while it strengthens its feedback, liquidity consumption enhancing effect. The end result is that traders aim to hedge the largest portion of their endowment, at the equilibrium where the cost of trading is at its highest. Importantly, we also find that: (i) depending on parameters values, uniqueness obtains either at an equilibrium with high or one with low liquidity (corresponding respectively to the high 4

5 and low liquidity equilibrium when multiplicity arises), and (ii) that the comparative statics properties of these equilibria differ. For instance, when the market hovers along an equilibrium with low liquidity, illiquidity can be hump-shaped in the proportion of fast dealers, something that does not happen in an equilibrium with high liquidity. The strategic complementarity loop arising with market opaqueness implies that liquidity can be fragile in our setup. We show this with two types of examples. In the first one, we exploit equilibrium multiplicity and illustrate how a small shock to some parameter values can produce a switch from the high liquidity equilibrium to an equilibrium with low liquidity. In particular, we focus on the consequence of a shock that disconnects a small mass of full dealers from the market (a technological glitch ). We then analyze the effect of a positive shock to the volatility of first and second period traders demand. These are meant to capture, respectively, an increase in the probability of a large order hitting the first period market (which is consistent with some narratives of the flash crash, see e.g. Easley et al. (20)), and an increase in the uncertainty first period traders face on their endowment value. In all these examples small parameter shocks produce large liquidity withdrawals. In the second type of example we review the impact of the glitch, but in this case leveraging on the hump-shaped relationship between illiquidity and full dealers participation that can obtain along an equilibrium with low liquidity. Based on this finding, we show that a high level of liquidity can suddenly evaporate because of a reduction in full dealers participation along the same equilibrium. This paper is related to four strands of the literature. First, equilibrium multiplicity, liquidity complementarities, and liquidity fragilities are known to obtain in economies where asset prices are driven by fundamentals information and noise trading (see, e.g., Cespa and Foucault (204), Cespa and Vives (205), Goldstein et al. (204), and Goldstein and Yang (205)). In this setup, in contrast, asset prices are exclusively driven by endowment shocks. However, the demand of all the traders is responsive to the volatility of the price at which these agents unwind their positions. In turn, such volatility depends on traders demand. As we argued above, in an opaque market this two-sided loop which in a noise traders economy cannot possibly arise is responsible for the multiplicity result. Other authors obtain multiple equilibria in setups where order flows are driven by only one type of shock (see, e.g., Spiegel (998)). However, multiplicity there arises from the bootstrap nature of expectations in the steady-state equilibrium of an overlapping generations (OLG) model in which investors live for two periods. Our setup, in contrast, considers an economy with a finite number of trading rounds. Second, the paper adds to the theoretical literature on the impact of high frequency trading (HFT) on market performance, by showing that an informational friction arising from liquidity provision fragmentation can be responsible for liquidity fragility, and reverses the common wisdom that associates an increase in computerized trading with more liquid markets. Differently from our setup, the HFT literature has mostly concentrated on modeling risk neutral agents (e.g., Budish et al. (205), Hoffmann (204), Du and Zhu (204), Bongaerts and Van Achter (205), Foucault et al. (205), and Menkveld and Zoican (205); see O Hara 5

6 (205) and Menkveld (206) for literature surveys). 6 Easley et al. (20, 202), find that in the hours preceding the flash crash, signed order imbalance for the E-mini S&P500 futures contract was unusually high. They interpret this evidence as supportive of a high order flow toxicity, which led HFTs to flee the market, eventually precipitating the crash. As argued above, our model also predicts that large imbalances can lead to a huge liquidity withdrawal. However, the channel we highlight is not related to adverse selection, but emphasizes the multiplier effect of illiquidity on the demand for immediacy that can arise when some traders have access to opaque information on imbalances. Menkveld and Yueshen (202) argue that market spatial fragmentation can be detrimental to stability. In their model, HFTs have access to a private reselling opportunity which, due to impaired intermarket connectivity, can break down. When this happens, HFTs trade among themselves, providing an illusion of liquidity to traders who observe volume, which in turn fosters further liquidity demand. Our focus is on the liquidity provision fragmentation induced by an informational friction in a single, concentrated market, a feature that is consistent with the futures markets flash events discussed above. Finally Han et al. (204), in a Glosten and Milgrom (985) setup with HFT and low frequency market makers, also find that illiquidity is hump-shaped in the proportion of HFT. However, in their model this result arises from an adverse problem that HFT s ability to cancel quotes upon the arrival of a negative signal creates for low frequency dealers. In our model, instead, dealers face no adverse selection risk, and illiquidity reversed U-shape arises because of strategic complementarities between liquidity consumption and provision. Third, the paper relates to the literature that assesses the impact of limited market participation. Heston et al. (200) and Bogousslavsky (204) find that some liquidity providers limited market participation can have implications for return predictability. Chien et al. (202) focus instead on the time-series properties of risk premium volatility. Hendershott et al. (204) concentrate on the effect of limited market participation for price departures from semi-strong efficiency. Our focus is, instead, on the destabilizing dynamics that is generated by bouts of illiquidity. In this respect, our paper is also related to Huang and Wang (2009) who show that with costly market participation, idiosyncratic endowment shocks can yield crashes. Note, however, that in our setup traders are exposed to the same shock, which yields a different mechanism for market instability. Fourth, by highlighting the first order asset pricing impact of uninformed traders imbalance predictability, this paper shares some features of our previous work (Cespa and Vives (202), and Cespa and Vives (205)). In that setup, however, predictability obtained because of the assumed statistical properties of noise traders demands, whereas in this paper it arises endogenously, because of a participation friction. A growing literature investigates the asset pricing implications of noise trading predictability. Collin-Dufresne and Vos (205) argue that informed traders time their entry to the presence of noise traders in the market. This, in turn, 6 Biais et al. (205) study the welfare implications of investment in the acquisition of HFT technology. In their model HFTs have a superior ability to match orders, and possess superior information compared to human (slow) traders. They find excessive incentives to invest in HFT technology, which, in view of the negative externality generated by HFT, can be welfare reducing. 6

7 implies that standard measures of liquidity (e.g., Kyle s lambda), may fail to pick up the presence of such traders. Peress and Schmidt (205) estimate the statistical properties of a noise trading process, finding support for the presence of serial correlation in demand shocks. The rest of the paper is organized as follows. In the next section we introduce the model, and show that with limited market participation, endowment shocks propagate across trading dates. Next, we analyze the benchmark with a transparent market. We then illustrate how the presence of an informational friction can generate strategic complementarities between traders demand for immediacy and market illiquidity. We show that such complementarities are at the root of the loop responsible for equilibrium multiplicity and liquidity fragility. A final section contains concluding remarks. All proofs are in the appendix. 2 The model A single risky asset with liquidation value v N(0, τ v ), and a risk-less asset with unit return are exchanged in a market during two trading rounds. Three classes of traders are in the market. First, a continuum of competitive, risk-averse, High Frequency Traders (which we refer to as Full Dealers and denote by FD) in the interval (0, µ), are active at both dates. Second, competitive, risk-averse dealers (D) in the interval [µ, ], are active only in the first period. Finally, a unit mass of short-term traders enters the market at date. At date 2, these traders unwind their position, and are replaced by a new cohort of short-term traders (of unit mass). The asset is liquidated at date 3. We now illustrate the preferences and orders of the different players. 2. Liquidity providers A FD has CARA preferences (we denote by γ his risk-tolerance coefficient) and submits pricecontingent orders x F t D, t =, 2, to maximize the expected utility of his final wealth: W F D = (v p 2 )x F 2 D + (p 2 p )x F D. 7 A Dealer also has CARA preferences with risk-tolerance γ, but is in the market only in the first period. He thus submits a price-contingent order x D to maximize the expected utility of his wealth W D = (v p )x D. The inability of D to trade in the second period captures some liquidity suppliers limited market participation. This friction could be due to technological reasons (as, e.g. in the case of standard dealers with impaired access to a technology that allows trading at high frequencies). 2.2 Short-term traders In the first period a unit mass of short-term traders is in the market. A short-term trader receives a random endowment of the risky asset u, and posts a market order x L anticipating that it will unwind its holdings in the following period, and leave the market. We assume 7 We assume, without loss of generality with CARA preferences, that the non-random endowment of FDs and dealers is zero. Also, as equilibrium strategies will be symmetric, we drop the subindex i. 7

8 u N(0, τ u ), and Cov[u, v] = 0. 8 First period traders have identical CARA preferences (we denote by γ L the common risk-tolerance coefficient). Formally, a trader maximizes the expected utility of his wealth π L = u p 2 + (p 2 p )x L : E [ exp{ π L /γ L } Ω L ], where Ω L denotes his information set. In period 2, first period traders are replaced by a new (unit) mass of traders receiving a random endowment of the risky asset u 2, where u 2 N(0, τ and Cov[u 2, v] = Cov[u 2, u ] = 0. u 2 ) A second period trader has CARA utility function with risk-tolerance γ L 2, and submits a market order to maximize the expected utility of his wealth π L 2 = u 2 v + (v p 2 )x L 2 : E [ exp{ π L 2 /γ L 2 } Ω L 2 ], where Ω L 2 denotes his information set. 9 We can interpret the second period traders as the proprietary desk of investment banks that trade to hedge their exposure to an asset whose payoff is perfectly correlated with the one of the asset traded in the market. 2.3 Information sets We now describe the information sets of the different market participants. At equilibrium, we conjecture that a period trader submits an order x L = b L u, where b L denotes the first period hedging aggressiveness, to be determined in equilibrium, while a FD and a dealer respectively post a limit order x F D = ϕ F D (p ), x D = ϕ D (p ) where ϕ F D ( ), ϕ D ( ) are linear functions of p. In the second period, a FD submits a limit order x F D 2 = ϕ 2 (p, p 2 ), where ϕ 2 ( ) is a linear function of prices. A second period trader observes a signal of the first period endowment shock s u = u + η, with η N(0, τ η ), and independent from all the other random variables in the model, and submits a market order x L 2 = b L 2u 2 + b L 22s u, where b L 2 and b L 22 denote respectively the second period hedging and speculative aggressiveness. With these assumptions, we obtain Lemma. At equilibrium, p is observationally equivalent to u, and the sequence {p, p 2 } is observationally equivalent to {u, x L 2 }. A first period trader observes the endowment shock u. Therefore, his information set coincides with the one of Ds and FDs: Ω L = Ω F D = Ω D = {u }. A second period trader receives an endowment shock u 2, and can observe a signal s u. Thus, his information set is Ω L 2 = {u 2, s u }. Finally, a FD in period 2 observes the sequence of prices: Ω F D 2 = {p, p 2 } from which he retrieves {u, x L 2 }. Thus, according to our model, liquidity provision is fragmented because (i) only one class of dealers is able to participate in the second period and (ii) some traders (the second cohort of 8 The assumption of a random endowment in the risky asset is akin to Huang and Wang (2009), and Vayanos and Wang (202) who instead posit that traders receive an endowment in a consumption good that is perfectly correlated with the value of the risky asset at the terminal date. 9 Our results are robust to the case in which the first period market is populated by a mass β of short-term traders, that unwind at date 2, and a mass ( β) of long-term ones that hold their position until liquidation. 8

9 short-term traders) have access to opaque information on the first period price. This assumption is consistent with the evidence that exchanges sell fuller access to their matching engine, as well as direct feeds of their market information at a premium (see, e.g., O Hara (205)). 0 Figure 2.3 displays the timeline of the model. Liquidity traders receive u and submit market order x L. FDs submit limit order µx F D. Dealers submit limit order ( µ)x D. 2 st period liquidity traders liquidate their positions. New cohort of liquidity traders receives u 2, observes s u, and submits market order x L 2. FDs submit limit order µx F D 2. 3 Asset liquidates. 2.4 Limited market participation and the propagation of endowment shocks Due to limited market participation, the first period endowment shock propagates to the second trading round, thereby affecting p 2. equation To see this, consider the first period market clearing µx F D + ( µ)x D + x L = 0. () At equilibrium the orders of first period traders are absorbed by both FDs and Ds. Thus, when µ <, FDs aggregate position falls short of x L : µx F D + x L 0. As a consequence, the inventory FDs carry over from the first period is insufficient to absorb the reverting orders that first period traders post in period 2. This creates an order imbalance driven by the first period endowment shock u that adds to the one originating from second period trades, and affects the second period price. Formally, from the second period market clearing equation we have µ(x F D 2 x F D ) + (x L 2 x L ) = 0. Substituting () in the latter and rearranging yields: µx F D 2 + x L 2 + ( µ)x D = 0. (2) 0 This assumption is also similar to Foucault et al. (205) who posit that HFTs receive market information slightly ahead of the rest of the market. Ding et al. (204) compare the NBBO (National Best Bid and Offer, which is the price feed computed by the Security Industry Processors in the US) to the fuller feeds market participants obtain via a direct access to different trading platforms. Their findings point to sizeable price differences that can yield substantial profits to HFTs. Latency in the reporting of market data can also be profitably exploited for securities with centralized trading, see High-speed traders exploit loophole, Wall Street Journal, May,

10 According to Lemma, at equilibrium x D the first period endowment shock. depends on u. Thus, when µ <, p 2 also reflects 2.5 Strategies We now discuss the strategies of the different market participants. In the second period, FDs act like in a static market: X F D 2 (p, p 2 ) = γτ v p 2. Therefore, they speculate on the asset payoff (recall that E[v] = 0), and supply liquidity, demanding a compensation that is inversely related to the risk they bear. In the first period, as we show in the appendix, we have X D (p ) = γ Var[v] p (3a) (p ) = γ E[p 2 p u ] γ Var[p 2 u ] Var[v] p }{{}}{{}. (3b) Speculation Market making X F D The above expressions imply that standard dealers accommodate the residual imbalance, while FDs also speculate on short term returns. The speculative component in FDs first period strategy has two implications. First, it makes the price adjustment FDs require to accommodate an increase in the aggregate demand for liquidity smaller compared to that required by Ds: ( ) X F D (p ; u ) = ( p γ Var[p 2 u ] + ) ( ) X D < (p ; u ) = Var[v]. (4) Var[v] p γ Second, it reduces the imbalance that liquidity providers (both Ds and FDs) have to clear at equilibrium. In particular, the larger is FDs speculative position, the smaller is the residual imbalance. Consider now short-term traders. In the appendix we show that a second period trader trades according to X2 L (u 2, s u ) = γ L E[v p 2 Ω L 2 ] 2 Cov[v p 2, v Ω L 2 ] u Var[v p 2 Ω L 2 ] Var[v p }{{} 2 Ω L 2 2 ] }{{} Speculation Hedging = γl 2 Cov[v p 2, u 2 ] Var[v p 2 Ω L 2 ]Var[u 2 ] u 2 }{{} Speculation on u 2 + γl 2 Cov[v p 2, s u ] Var[v p 2 Ω L 2 ]Var[s u ] s u } {{ } Speculation on u (5) Cov[v p 2, v Ω L 2 ] u Var[v p 2 Ω L 2 2 ] }{{} Hedging (6) Thus, a trader s strategy has a speculative and a hedging component. According to the first line in (5), a trader speculates on value change the more, the less liquid is the market (see the first term on the r.h.s. in (5)), while lowering his exposure to the asset risk the more, the higher is the covariance between the return on his position (i.e., v p 2 ) and the final liquidation value 0

11 (v), given his information. In this way he reduces the risk that his speculative strategy goes sour precisely when the value of his endowment collapses. Expanding the expectation operator at the numerator of (5) shows that there are two sources of speculation. Other things equal, given u 2 a trader retains part of his asset exposure to the extent that this is positively correlated with the capital gain v p 2, to profit from the latter. Additionally, he uses his information on u to speculate on the reverting orders of first period traders. Thus, an alternative interpretation for these traders is that they are akin to the opportunistic traders of the CFTC-SEC report taxonomy. Indeed, according to the report, these agents accumulate directional long or short position(s) in a way that is consistent with different arbitrage strategies. First period traders strategies are similar to (5): X L (u ) = γ L E[p 2 p u ] Var[p 2 u ] } {{ } Speculation on u u }{{}. (7) Hedging First period traders can partially anticipate the second period price, and thus speculate on it, for example by holding part of their endowment when u > 0. Substituting (7) in (3b) yields the following expression: X F D (p ) = γ γ L ( X L (u ) + u ) γ Var[v] p. (8) According to (8), for given u, a contraction of first period traders holdings (i.e., an increase in their demand for liquidity), leads to a corresponding contraction in FDs speculative activity, and thus to an increase in the residual imbalance that liquidity suppliers have to clear in equilibrium. 3 Market transparency and the rationing effect of illiquidity In this section, we assume that second period traders have a perfect signal on the first period endowment shock: τ η. This captures a scenario in which information on the first period imbalance is public, as is the case in a low frequency trade environment (e.g., intradaily). Alternatively, it represents an ideal setup in which second period traders have access to the same information as FDs. In this case, we obtain the following result: Proposition. When the market is transparent there exists a unique equilibrium in linear strategies, where x L = b L u, x L 2 = b L 2u 2 + b L 22s u, p 2 = λ 2 (b L 2u 2 + b L 22s u ) + λ 2 ( µ)γτ v Λ u p = Λ u, (9a) (9b)

12 λ 2 = /(µγτ v ) > 0, and b L = γ L Λ = γτ v Cov[p 2, u ]τ u + Λ Var[p 2 u ] ( (µγ + γl )( + b L ) γ L b L 2 = µγ µγ + γ L 2 ) (, µγ ) µγ + γ L (0a) (0b) (0c) b L 22 = γl 2 b L 2( µ)λ τ v. (0d) µ The coefficient Λ, i.e. the first period endowment shock s negative price impact, is our measure of liquidity: Λ = p u. () As is standard in economies with noise traders and risk-averse liquidity suppliers, Λ reflects dealers compensation to absorb the outstanding imbalance in their inventory: the cost of supplying liquidity. However, differently from a noise trader economy, in this model dealers inventory depends on the equilibrium trading decisions of FDs and first period traders. To see this, consider (0a). In view of (7) and (0b), at equilibrium first period traders hold a fraction + b L = γ L Cov[p 2, u ]τ u + Λ, (2) Var[p 2 u ] of their endowment shock. At the same time, comparing (3b) with (7), FDs aggregate speculative position per unit of endowment shock is given by µγ E[p 2 p u ] = µγ + bl. (3) Var[p 2 p ]u γ L Thus, summing (2) and (3) yields the total speculative exposure of FDs and first period traders per unit of u (i.e., the fraction of the endowment shock that is not absorbed by liquidity suppliers): + b L + µγ + bl = (µγ + γl )( + b L ), (4) γ L γ L and the complement to one of (4) captures dealers inventory (per unit of endowment shock): Dealer s inventory per unit of endowment shock = (µγ + γl )( + b L ). (5) γ L At date FDs know that they will be able to unwind their inventory in the second trading round, when x L reverts. However, at that point in time, a new generation of traders enters the market. These traders hedge a new endowment shock, exposing FDs to the risk of holding their initial inventory until the liquidation date. Thus, for given inventory (5), the riskier is the asset, and the more risk averse FDs are, the higher is the risk borne by liquidity suppliers, and, according to (0a), the less liquid is the market. 2

13 According to (0b) and (0c), first and second period traders demand liquidity to hedge a fraction of their endowment. In the second period, such a fraction corresponds to FDs relative risk-bearing capacity (see (0c)); in the first period, instead, it is larger than that (see (0b)). This is because second period traders hedging activity creates price volatility which heightens first period traders uncertainty, and leads them to demand more liquidity and FDs to cut back on their speculative activity. Indeed, when second period traders endowment shock is null, first period traders hedging aggressiveness reaches its upper bound, and the market is infinitely liquid: Corollary. In a transparent market, when the second period endowment shock is null (τ u2 ), first period traders liquidity demand matches FDs relative risk-bearing capacity, and the market is infinitely liquid (b L (µγ + γ L ) µγ, and Λ 0). According to (0d), second period traders also speculate on the propagated order imbalance by putting a negative weight on their signal (b L 22 < 0), which is increasing in Λ. This is because, for u > 0, the reversion of first period trades creates a positive imbalance at date 2, which prompts second period traders to short the asset. A less liquid first period market makes it more profitable for Ds to absorb u, which strengthens the positive dependence between p 2, and u since then less of the first period s traders reversion in period 2 is accomodated by FDs: Cov[p 2, u ] = ( µ)λ 2τ v Λ τ u ( ) γ L 2 b L 2 µ + γ. (6) Thus, as Λ increases, second period traders step up their speculative aggressiveness. The equilibrium (Λ, bl) obtains as the unique solution to the system (0a) (0b): Λ = ( (µγ + ) γl )( + b L ) γτ v γ L b L = γ L (γ + γ L 2 )(µγ + γ L 2 )τ 2 vτ u2 Λ, (7a) (7b) which can be understood as the intersection between the inverse supply and demand of liquidity (respectively, (7a) and (7b)). This is so because b L measures the fraction of the endowment shock that first period traders hedge in the market, while Λ captures the price adjustment dealers require to accommodate the order imbalance. A less liquid first period market increases the cost of scaling down traders exposure, and leads the latter to hedge less of their endowment. Thus, in this case a drop in liquidity has a rationing effect on liquidity consumption, and the demand for liquidity is a decreasing function of Λ. Conversely, a lower hedging aggressiveness implies a larger speculative position for FDs, which shrinks the imbalance that liquidity suppliers have to clear in the first period, and leads to a more liquid market. Hence, the (inverse) supply of liquidity is decreasing in b L. In Figure we provide a graphical illustration of the equilibrium determination. As b L < 0, and positively sloped in Λ, a higher illiquidity implies that traders shed a lower fraction of their endowment, or that their liquidity demand subsides. 3

14 bl 0.0 γ=, γl=/2, γ2l=, τu=/0, τu2=200, τv=/ Λ Λ bl Figure : Transparency and equilibrium uniqueness. The solid (dashed) curves are drawn assuming µ = /0 (µ = /5). When µ = /0, {Λ, b L } = {.4,.2}, while when µ = /5, {Λ, b L } = {.3,.3}. In Figure we also graphically analyze the effect of an increase in the mass of FDs on Λ. The solid (dashed) curves in the figure are drawn for µ = /0 (µ = /5). A larger µ has a positive effect on the cost of trading for all levels of b L, since, according to (3), the aggregate speculative position of FDs increases, lowering dealers inventory. As a result, when µ increases, the new function Λ shifts downwards. Consider now b L. Based on (2), a larger µ has two contrasting effects on first period traders hedging aggressiveness: on the one hand, as one can compute using (9a) and (9b), first period return uncertainty is given by: Var[p 2 u ] = (λ 2b L 2) 2 τ u2 = (µγ + γ L 2 ) 2 τ 2 vτ u2, (8) which is decreasing in µ. Therefore, a larger µ lowers first period traders uncertainty about p 2, and makes them consume less liquidity. However, according to (6), Cov[p 2, u ] µ < 0 (9) and a higher µ lowers the positive association between the second period price and the first period endowment shock, making speculation less profitable. This pushes first period traders to shed a larger fraction of their endowment, increasing dealers inventory, and consuming more liquidity. When the market is transparent, this latter effect is never strong enough to offset the former two and we obtain: Corollary 2. In a transparent market, liquidity increases in the proportion of fast dealers ( Λ / µ < 0). 4

15 We concentrate our analysis on the liquidity of the first period market. However, note that as the volatility of the first period price is given by Var[p ] = (Λ ) 2 τ u, our liquidity results can also be interpreted in terms of price volatility. 4 Opaqueness and the feedback effect of illiquidity Suppose now that second period traders signal on u has a bounded precision (τ η < ). This setup characterizes a scenario where some traders (FDs, in our setup) have access to better market information (for example on order imbalances) compared to others (the second cohort of traders), and given our previous discussion, is likely to hold at a high trading frequency. In this case, we obtain the following result: Proposition 2. When 0 < τ η <, at equilibrium: x L = b L u, x L 2 = b L 2u 2 + b L 22s u, b L 2 = µγ γ L 2 + µγκ b L 22 = γ L 2 b L 2τ v τ η Cov[p 2, u Ω L 2 ], (20a) (20b) where κ τ v Var[v p 2 Ω L 2 ] >, (2a) and the first and second period return uncertainty are respectively given by Var[p 2 u ] = λ 2 2((b L 2) 2 /τ u2 + (b L 22) 2 /τ η ), and Var[v p 2 Ω L 2 ] = Var[v] + (λ 2 ( µ)γτ v Λ ) 2 Var[u s u ]. Differently from the transparent market benchmark, second period traders now face uncertainty on the price at which their order is executed, besides that on the liquidation value. This additional source of uncertainty is captured by the coefficient κ (see (2a)). As a consequence, they hedge a lower fraction of their endowment shock (see (20a)). Other things equal, as µ increases, u propagates less to period 2, κ tends to, and second period traders (i) hedge more of their endowment shock, and (ii) speculate less aggressively on the propagated shock: lim µ bl 2 = γ γ L 2 + γ, lim µ bl 22 = 0. (22) We are now ready to analyze the effect of a shock to liquidity on the equilibrium coefficients: Corollary 3. At equilibrium, the impact of the first period endowment shock on the second period price, second period traders return uncertainty and hedging aggressiveness are increasing in illiquidity: Cov[p 2, u ] Λ > 0, Var[v p 2 Ω L 2 ] Λ > 0, b L 2 Λ > 0. (23) An increase in Λ has an ambiguous effect on first period traders return uncertainty, and on second period traders speculative aggressiveness (Var[p 2 u ], and b L 22). 5

16 According to (23) as in the transparent market case, a less liquid first period market increases the positive association between p 2 and u. Furthermore, as second period traders do not perfectly observe u, this also augments these traders uncertainty and, according to (23), lowers their hedging responsiveness (recall that b L 2 < 0). Importantly, a higher illiquidity has two contrasting effects on the speculative aggressiveness of second period traders (b L 22). Direct computation yields: Cov[p 2, u Ω L 2 ] = λ 2 ( µ)γτ v Λ Var[u s u ]. Thus, differentiating b L 22 we obtain: b L 22 Λ = γ L 2 τ v τ η Cov[p 2, u Ω L 2 ] bl 2 + b L 2 Λ }{{ } Uncertainty effect (+) Cov[p 2, u Ω L 2 ] Λ } {{ } Speculation effect ( ). (24) On the one hand, like in the transparent market benchmark, an increase in Λ augments second period traders speculative opportunities, and drives them to trade more against the u -led imbalance (the second term in the parenthesis in (24)). On the other hand, a higher Λ augments second period traders return uncertainty, and makes them speculate less (the first term in the parenthesis). Consider now the effect of an increase in Λ on Var[p 2 u ]: Var[p 2 u ] = 2λ 2 b L 2 b L 2 2 Λ τ u2 Λ }{{ } ( ) b L 22 τ η Λ }{{. (25) } (±) + bl 22 In the transparent market benchmark, a higher illiquidity has no impact on first period traders uncertainty over p 2 (see (8)). In contrast, according to (23), opaqueness introduces two channels through which a shock to liquidity feeds back to first period traders uncertainty. First, an increase in Λ lowers second period traders hedging activity, lowering Var[p 2 u ]. However, as we argued above, a less liquid first period market can spur more speculation by second period traders. As traders information is imprecise, this yields a second feedback channel that can instead magnify first period traders uncertainty. Thus, according to (25), the ultimate impact of a shock to Λ on first period traders uncertainty depends on the strength of the speculation effect. Finally, because of opaqueness, an increase in Λ introduces an additional effect on b L : b L Λ = γ L Var[p 2 u ] (26) 2 ( ) Cov[p2, u ] τ u + Var[p 2 u ] Var[p 2 u ] (Cov[p 2, u ]τ u + Λ ) Λ } Λ {{}}. {{} Direct effect (+) Feedback effect (±) For given Var[p 2 u ], as p 2 is more positively associated with u, a larger Λ leads first period traders to speculate more (and hedge less), as per the direct liquidity consumption rationing effect of the transparent market benchmark. However, when the speculation effect 6

17 leads Var[p 2 u ] to increase in Λ, a less liquid market now also has a feedback liquidity consumption expanding effect on b L. As a higher Λ increases the risk to which first period traders are exposed, a less liquid market can lead them to hedge more. As a result, first period traders demand for liquidity can become increasing in Λ, as shown in Figure 2: an increase in the cost of liquidity provision incites more liquidity consumption. The expanding effect of illiquidity can be responsible for a destabilizing dynamic whereby to a sizeable evaporation of liquidity, first period traders respond with an even more aggressive liquidity consumption. In the figure we use the same parameter values of Figure, but assume that τ η = 0 (instead of τ η ). As a result, at equilibrium we obtain b L τ η=0 = 0.5, Λ τ η=0 = 3.8. Compared to the values of the example of Figure, these results correspond to a more than two- and an almost ten-fold increase in liquidity consumption and illiquidity. γ=, γl=/2, γ2l=, τu=/0, τu2=200, τv=/0, τη=0, μ=/0 bl Λ Λ, Opaque bl, Opaque bl, Transparent -2.0 Figure 2: When the market is opaque, first period traders demand for liquidity can turn increasing in Λ. The dashed curve corresponds to b L in the transparent market case. 4. Equilibrium multiplicity A second effect of opaqueness is the possibility of multiple, self-fulfilling equilibria which arise out of strategic complementarities in liquidity demand. According to Corollary 3, a less liquid first period market heightens the time-propagation of the first period shock. This, in turn, can lead second period traders to speculate more aggressively on the u -led imbalance (see (24)), which can increase the uncertainty faced by first period traders on p 2 (see (25)). As a consequence, first period traders can decide to hedge more, and FDs to speculate less (see (26)). 2 This chain of effects turns out to be particularly strong when the risk bearing capacity of FDs is not too low, first period traders are sufficiently risk averse, second period traders have a sufficiently informative signal, and face low endowment risk, and the risk of the asset payoff is large. In these conditions, an initial dearth of liquidity escalates into a loop that sustains three equilibrium levels of liquidity: 2 Because of (3), whenever first period traders consume more liquidity, FDs speculate less, increasing the inventory held by liquidity suppliers. 7

18 Proposition 3. There exists a set of parameter values {τ u2, τ v, τ η, µ, γ, γ L }, such that for τ u2 > τ u2, τ v < τ v, τ η > τ η, µ < µ, γ > γ, and γ L < γ L, three equilibrium levels of liquidity (Λ ) H, (Λ ) I, (Λ ) L arise, where 0 < (Λ ) H < µ µ < (Λ ) I < µ < (Λ ) L < γτ v. (27) We will refer to the equilibrium where Λ is low (resp., intermediate, and high) as the High, (resp., Intermediate, and Low) liquidity equilibrium (HLE, ILE, and LLE). Note that since the function Λ (b L ) is decreasing in b L (see (7a) ), the hedging activity of first period traders is respectively high, intermediate, and low along (Λ ) L, (Λ ) I, and (Λ ) H. This is a further manifestation of the fact that the feedback effect of liquidity jams the stabilizing impact of an increase in illiquidity on traders hedging demand. We can interpret the ratios µ µ, γτ v, (28) in (27), respectively, as the likelihood that FDs second period liquidity supply is enough to absorb the demand coming from first period traders reverting orders, and as liquidity suppliers perceived uncertainty about the asset payoff. Then, condition (27) states that when multiplicity arises, the likelihood that FDs inventory is sufficient to stave off a liquidity shortage is smaller than FDs perceived asset payoff risk. Figure 3 provides a numerical example of the proposition. γ=9/0, γl=/5, γ2l=9/0, τu=2, τu2=600, τv=/0, τη=0, μ=/5 bl Λ Λ, Opaque Λ * Λ * bl, Opaque bl, Transparent Figure 3: Market opaqueness and equilibrium multiplicity. At equilibrium {Λ, b L } {{0.4, 0.5}, {, 0.5}, {3.9, 0.7}}. The following corollary follows from Proposition 3: Corollary 4. When the volatility of the second period endowment shock vanishes (τ u2 ) and the following parameter restriction applies: τ v < τ v, τ η > τ η, µ < µ, γ L < γ L, then (Λ ) H = 0. When τ u2, second period traders have no endowment to hedge, and only trade to speculate on the u -induced imbalance. In the equilibrium where Λ = 0, x D = 0, so that first period traders orders are absorbed by FDs speculative trades, no imbalance arises in the second period, and b L 22 = 0 (see (20b)). When second period traders signal on u is fully revealing, this equilibrium is unique (Corollary ). For τ η finite, however, first period traders 8

19 cannot rule out the possibility that second period traders speculate on a certain realization of s u that gives an incorrect signal about u (e.g., s u > 0, while u < 0). This increases the uncertainty they face, and trigger the loop that can lead to the appearance of two further equilibria. Figure 4 provides a graphical illustration of the equilibrium determination when the conditions in Corollary 4 are satisfied. bl γ=9/0, γl=/5, γ2l=9/0, τu=2, τv=/0, τη=0, μ=/ Λ Λ, Opaque bl, Opaque -2.0 Figure 4: Equilibrium multiplicity with no second period endowment risk. At equilibrium {Λ, b L } {{0, 0.5}, {., 0.5}, {4.2, 0.7}} (the function b L is very steep at Λ = 0). To study comparative statics, it is convenient to introduce the equilibrium best response function ψ(λ ), (29) which collapses the demand and supply of liquidity equations into a single one that can be interpreted as an aggregate best response of first period traders to a change in first period illiquidity. As we explain in the appendix, the fixed points of (29) correspond to the equilibria of the market. 3 Figure 5 provides a graphical representation of the best response, for a set of parameters yielding multiple equilibria. γ=9/0, γl=/5, γ2l=9/0, τu=2, τu2=400, τη=0, τv=/0, μ=0.2 ψ(λ ) 5 4 (Λ * ) L = (Λ * ) I =.03 (Λ * ) H = Λ Figure 5: The best response mapping (29) for γ = γ L 2 = 9/0, γ L = /5, τ u = 2, τ u2 = 400, τ η = 0, τ v = /0, and µ = /5. 3 See the proof of Proposition 2, and in particular the discussion around (A.27). 9

20 According to our analysis, a positive shock to τ u2, γ L 2, and τ η, (for intermediate values of the signal precision), tends to strengthen the degree of strategic complementarity, facilitating equilibrium multiplicity. Intuitively, in these conditions second period traders are relatively better positioned to speculate on the propagated imbalance, which reinforces the impact of their orders on first period traders uncertainty. A positive shock to µ, γ, γ L, τ u, and τ v moves the best response down and also mildly decreases the degree of strategic complementarity. These shocks increase the risk bearing capacity of the market, and reduce the risk to which first period traders are exposed. As a consequence, they weaken the impact of second period traders orders on first period traders uncertainty More in detail, as shown in Figures and 2:. An increase in µ increases the risk bearing capacity of the market, and lowers the second period imbalance due to u. This shifts the best response mapping downwards, implying a higher (lower) liquidity at (Λ ) H, (Λ ) L ((Λ ) I ). A similar effect obtains when γ or τ v increase (see Figure, panel (a), (b), and (c)). 2. An increase in γ L or in τ u works instead to lower the supply shock that Ds and FDs absorb in the first period, and thus the second period imbalance due to u, shifting the best response mapping downwards (see Figure, panel (d) and (e)). 3. An increase in τ η has two contrasting effects on the strength of the loop. For τ η small, a more precise signal on u boosts second period traders speculation on the u -induced imbalance, heightening first period traders uncertainty on p 2, increasing Λ, and strengthening the loop. As τ η increases further, Var[p 2 u ] starts decreasing (see the expression in Proposition 2), leading first period traders to hold more of their endowment shock, increasing the liquidity of the first period market, reducing the size of the u -led imbalance, and weakening the loop. When τ η, (i) the impact of second period traders speculation on Var[p 2 u ] disappears, and (ii) second period traders uncertainty no longer depends on Λ (see the expression for Var[v p 2 Ω L 2 ] in Proposition 2). This severs the link between trading decisions at the two dates, yielding a unique equilibrium (as we know from the analysis of Section 3). Figure 2, Panel (a), illustrates this effect. Note that as τ η increases, the intermediate and low liquidity equilibria eventually disappear, but liquidity at the high liquidity equilibrium diminishes (compared with the case with low signal precision). 4. To understand the effect of a change in τ u2, recall that according to Corollary 4 in the extreme case in which the second period endowment shock is null (almost surely), there always exist an equilibrium where Λ = 0, x D = 0, so that first period traders orders are absorbed by FDs speculative trades, which implies that no imbalance arises in the second period, and b L 22 = 0 (see (20b)). 4 When second period traders signal on u is fully revealing, this equilibrium is unique. For τ η finite, however, first period traders cannot 4 A full analytical characterization of this equilibrium is complex. Numerically, it can be seen that first period liquidity traders hedge the smallest possible fraction of their endowment shock: b L µγ/(µγ + γ L ). 20